An alternative approach, known today as the Bernstein polynomials, to Weierstrass uniform approximation theorem was provided by Bernstein. These basis polynomials have attained increasing momentum, especially in operator theory, integral equations and computer-aided geometric design. Motivated improvements of computational disciplines, we propose a new generalization Bernstein–Kantorovich opera...

This paper describes an algorithm that can be used to effectively solve polynomial constraints over finite domains. Such constraints are expressed in terms of inequalities of polynomials with integer coefficients whose variables are assumed to be defined over proper finite domains. The proposed algorithm first reduces each constraint to a canonical form, i.e., a specific form of inequality, the...

We show that the classical Bernstein polynomials BN(f)(x) on the interval [0, 1] (and their higher dimensional generalizations on the simplex Σm ⊂ R) may be expressed in terms of Bergman kernels for the Fubini-Study metric on CP: BN(f)(x) is obtained by applying the Toeplitz operator f(N−1Dθ) to the Fubini-Study Bergman kernels. The expression generalizes immediately to any toric Kähler variety...

We show that the classical Bernstein polynomials BN (f)(x) on the interval [0, 1] (and their higher dimensional generalizations on the simplex Σm ⊂ R) may be expressed in terms of Bergman kernels for the Fubini-Study metric on CP: BN (f)(x) is obtained by applying the Toeplitz operator f(N−1Dθ) to the Fubini-Study Bergman kernels. The expression generalizes immediately to any toric Kähler varie...

One of the standard problems in statistics consists of determining the relationship between a response variable and a single predictor variable through a regression function. Background scientific knowledge is often available that suggests the regression function should have a certain shape (e.g., monotonically increasing or concave) but not necessarily a specific parametric form. Bernstein pol...

Then, for each point x of continuity of f , we have Bn(f )(x)→ f (x) as n→∞. Moreover, if f is continuous on [, ] then Bn(f ) converges uniformly to f as n→∞. Also, for each point x of differentiability of f , we have B′n(f )(x)→ f ′(x) as n→∞ and if f is continuously differentiable on [, ] then B′n(f ) converges to f ′ uniformly as n→∞. Bernstein polynomials have been generalized in the fr...

This paper presents two main results. The first result pertains to uniform approximation with Bernstein polynomials. We show that, given a power-form polynomial g, we can obtain a Bernstein polynomial of degree m with coefficients that are as close as desired to the corresponding values of g evaluated at the points 0, 1 m , . . . , 1, provided that m is sufficiently large. The second result per...